Internal problem ID [3797]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 36
Problem number: 1092.
ODE order: 1.
ODE degree: 4.
CAS Maple gives this as type [[_homogeneous, `class G`]]
\[ \boxed {{y^{\prime }}^{4}-4 x^{2} y {y^{\prime }}^{2}+16 x y^{2} y^{\prime }-16 y^{3}=0} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 122
dsolve(diff(y(x),x)^4-4*x^2*y(x)*diff(y(x),x)^2+16*x*y(x)^2*diff(y(x),x)-16*y(x)^3 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {x^{4}}{16} \\ y \left (x \right ) = 0 \\ y \left (x \right ) \left (\sqrt {x^{2}-4 \sqrt {y \left (x \right )}}+x \right )^{\frac {2 \sqrt {y \left (x \right ) x^{2}-4 y \left (x \right )^{\frac {3}{2}}}}{\sqrt {x^{2}-4 \sqrt {y \left (x \right )}}\, \sqrt {y \left (x \right )}}} \left (\sqrt {x^{2}-4 \sqrt {y \left (x \right )}}-x \right )^{-\frac {2 \sqrt {y \left (x \right ) x^{2}-4 y \left (x \right )^{\frac {3}{2}}}}{\sqrt {x^{2}-4 \sqrt {y \left (x \right )}}\, \sqrt {y \left (x \right )}}}-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 32.07 (sec). Leaf size: 519
DSolve[(y'[x])^4 -4 x^2 y[x] (y'[x])^2+16 x y[x]^2 y'[x]-16 y[x]^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {\sqrt {\left (x^2+4 \sqrt {y(x)}\right ) y(x)} \log \left (\sqrt {x^2+4 \sqrt {y(x)}}-x\right )}{\sqrt {x^2+4 \sqrt {y(x)}} \sqrt {y(x)}}+\frac {1}{4} \left (\log (y(x))-\frac {\sqrt {x^2+4 \sqrt {y(x)}} \sqrt {y(x)} \log (y(x))}{\sqrt {\left (x^2+4 \sqrt {y(x)}\right ) y(x)}}\right )=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {1}{4} \left (\frac {\sqrt {x^2+4 \sqrt {y(x)}} \sqrt {y(x)} \log (y(x))}{\sqrt {\left (x^2+4 \sqrt {y(x)}\right ) y(x)}}+\log (y(x))\right )-\frac {\sqrt {\left (x^2+4 \sqrt {y(x)}\right ) y(x)} \log \left (\sqrt {x^2+4 \sqrt {y(x)}}-x\right )}{\sqrt {x^2+4 \sqrt {y(x)}} \sqrt {y(x)}}=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {1}{2} \left (\frac {\sqrt {x^2-4 \sqrt {y(x)}} \sqrt {y(x)} \log (y(x))}{2 \sqrt {x^2 y(x)-4 y(x)^{3/2}}}+\frac {1}{2} \log (y(x))\right )-\frac {\sqrt {\left (x^2-4 \sqrt {y(x)}\right ) y(x)} \log \left (\sqrt {x^2-4 \sqrt {y(x)}}-x\right )}{\sqrt {x^2-4 \sqrt {y(x)}} \sqrt {y(x)}}=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {\sqrt {\left (x^2-4 \sqrt {y(x)}\right ) y(x)} \log \left (\sqrt {x^2-4 \sqrt {y(x)}}-x\right )}{\sqrt {x^2-4 \sqrt {y(x)}} \sqrt {y(x)}}+\left (\frac {1}{4}-\frac {\sqrt {x^2 y(x)-4 y(x)^{3/2}}}{4 \sqrt {x^2-4 \sqrt {y(x)}} \sqrt {y(x)}}\right ) \log (y(x))=c_1,y(x)\right ] \\ y(x)\to 0 \\ y(x)\to \frac {x^4}{16} \\ \end{align*}